Subgroup Test (one-step)

Subgroup Test (one-step): Let $G$ be a group and let $H$ be a nonempty subset of $G$. Then $H$ is a subgroup of $G$ if and only if for all $x,y\in H$ one has $xy^{-1}\in H$.

This is often the fastest criterion to check the subgroup property because it packages “closed under products” and “closed under inverses” into a single closure condition.

Proof sketch: If $H$ is a subgroup, then $y^{-1}\in H$ and hence $xy^{-1}\in H$. Conversely, assume $H\neq\varnothing$ and $xy^{-1}\in H$ for all $x,y\in H$. Fix $h\in H$. Taking $x=y=h$ gives $e=hh^{-1}\in H$. Taking $x=e$ shows $y^{-1}\in H$ for all $y\in H$, and then $xy=(x)( (y^{-1})^{-1})\in H$ shows closure under products.